Solved: An article in the Tappi Journal (March, 1986) presented data on green liquor

An article in Concrete Research (Near Surface Characteristics of Concrete: Intrinsic Permeability, Vol. 41, 1989), presented data on compressive strength x and intrinsic permeability y of various concrete mixes and cures. Summary quantities are n  14, gyi  572, g  23,530, g xi  43,  157.42, and g xiyi  1697.80. Assume that the two variables are related according to the simple linear regression model. (a) Calculate the least squares estimates of the slope and intercept. (b) Use the equation of the fitted line to predict what permeability would be observed when the compressive strength is x  4.3. (c) Give a point estimate of the mean permeability when compressive strength is x  3.7. (d) Suppose that the observed value of permeability at x  3.7 is y

Regression methods were used to analyze the data from a study investigating the relationship between roadway surface temperature (x) and pavement deflection ( y). Summary quantities were n  20, g yi  12.75,  8.86, g xi  1478, gx  143,215.8, and g xiyi  1083.67.

11-3. Consider the regression model developed in Exercise 11-2. (a) Suppose that temperature is measured in C rather than F. Write the new regression model that results. (b) What change in expected pavement deflection is associated with a 1 C change in surface temperature?

Montgomery, Peck, and Vining (2001) present data concerning the performance of the 28 National Football League teams in 1976. It is suspected that the number of games won (y) is related to the number of yards gained rushing by an opponent (x). The data are shown in the following table.

An article in Technometrics by S. C. Narula and J. F. Wellington (Prediction, Linear Regression, and a Minimum Sum of Relative Errors, Vol. 19, 1977) presents data on the selling price and annual taxes for 24 houses. The data are shown in the following table. (a) Assuming that a simple linear regression model is appropriate, obtain the least squares fit relating selling price to taxes paid. What is the estimate of 2 ?

The number of pounds of steam used per month by a chemical plant is thought to be related to the average ambient temperature (in F) for that month. The past years usage and temperature are shown in the following table:

11-7. The data shown in the following table are highway gasoline mileage performance and engine displacement for a sample of 20 cars.

An article in the Tappi Journal (March, 1986) presented data on green liquor Na2S concentration (in grams per liter) and paper machine production (in tons per day). The data (read from a graph) are shown as follows: y 40 42 49 46 44 48 x 825 830 890 895 890 910 y 46 43 53 52 54 57 58 x 915 960 990 1010 1012 1030 1050 (a) Fit a simple linear regression model with y  green liquor Na2S concentration and x  production. Find an estimate of 2 . Draw a scatter diagram of the data and the resulting least squares fitted model. (b) Find the fitted value of y corresponding to x  910 and the associated residual. (c) Find the mean green liquor Na2S concentration when the production rate is 950 tons per day.

11-9. An article in the Journal of Sound and Vibration (Vol. 151, 1991, pp. 383394) described a study investigating the relationship between noise exposure and hypertension. The following data are representative of those reported in the article. y 1012514 6 2 3 x 60 63 65 70 70 70 80 90 80 80 y 5468457 9 7 6 x 85 89 90 90 90 90 94 100 100 100 (a) Draw a scatter diagram of y (blood pressure rise in millimeters of mercury) versus x (sound pressure level in decibels). Does a simple linear regression model seem reasonable in this situation? (b) Fit the simple linear regression model using least squares. Find an estimate of 2 . (c) Find the predicted mean rise in blood pressure level associated with a sound pressure level of 85 decibals.

An article in Wear (Vol. 152, 1992, pp. 171181) presents data on the fretting wear of mild steel and oil viscosity. Representative data follow, with x  oil viscosity and y  wear volume ( cubic millimeters).

11-11. An article in the Journal of Environmental Engineering (Vol. 115, No. 3, 1989, pp. 608619) reported the results of a study on the occurrence of sodium and chloride in surface streams in central Rhode Island. The following data are chloride concentration y (in milligrams per liter) and roadway area in the watershed x (in percentage). y 4.4 6.6 9.7 10.6 10.8 10.9 x 0.19 0.15 0.57 0.70 0.67 0.63 y 11.8 12.1 14.3 14.7 15.0 17.3 x 0.47 0.70 0.60 0.78 0.81 0.78 y 19.2 23.1 27.4 27.7 31.8 39.5 x 0.69 1.30 1.05 1.06 1.74 1.62 (a) Draw a scatter diagram of the data. Does a simple linear regression model seem appropriate here? (b) Fit the simple linear regression model using the method of least squares. Find an estimate of 2 . (c) Estimate the mean chloride concentration for a watershed that has 1% roadway area. (d) Find the fitted value corresponding to x  0.47 and the associated residual.

A rocket motor is manufactured by bonding together two types of propellants, an igniter and a sustainer. The shear strength of the bond y is thought to be a linear function of the age of the propellant x when the motor is cast. Twenty observations are shown in the table on the next page. (a) Draw a scatter diagram of the data. Does the straight-line regression model seem to be plausible? (b) Find the least squares estimates of the slope and intercept in the simple linear regression model. Find an estimate of 2 . (c) Estimate the mean shear strength of a motor made from propellant that is 20 weeks old. (d) Obtain the fitted values that correspond to each observed value yi . Plot versus yi , and comment on what this plot would look like if the linear relationship between

Show that in a simple linear regression model the point ( ) lies exactly on the least squares regression line.

Consider the simple linear regression model Y  0  1x  . Suppose that the analyst wants to use z  x as the regressor variable. (a) Using the data in Exercise 11-12, construct one scatter plot of the ( ) points and then another of the ( ) points. Use the two plots to intuitively explain how the two models, Y  0  1x  and , are related. (b) Find the least squares estimates of and in the model . How do they relate to the least squares estimates and ?

Suppose we wish to fit the model  , where (i  1, 2, p , n). Find the least squares estimates of and . How do they relate to and ?

Suppose we wish to fit a regression model for which the true regression line passes through the point (0, 0). The appropriate model is Y  x  . Assume that we have n pairs of data (x1, y1), (x2, y2), p , (xn, yn). Find the least squares estimate of .

11-17. Using the results of Exercise 11-16, fit the model Y  x  to the chloride concentration-roadway area data in Exercise 11-11. Plot the fitted model on a scatter diagram of the data and comment on the appropriateness of the model.